Unbiased Subsurface Flow and Transport Estimators Are Non-Darcian and Non-Fickian

Abstract:

Subsurface solute flow and transport take place in a complex heterogeneous environment. Neither the underlying flow and transport phenomena nor their host environment can be observed or described with certainty at all scales and locations of relevance. It has thus become increasingly common to represent hydraulic and transport characteristics of geologic media by random functions of space and to model flow and transport in such media stochastically. One approach is to generate multiple random realizations of medium hydraulic and transport properties, primarily permeability and (less commonly) advective porosity, conditional on measured values of these quantities (when available) using geostatistical techniques; solving standard Darcian flow and Fickian transport equations in each conditionally generated random property field; and averaging the results. Quantities averaged in this manner constitute smooth but optimal estimates of their true but unknown counterparts. Significantly, fluid and solute fluxes estimated in this manner are respectively non-Darcian and non-Fickian. This is reflected unambiguously in the space-time nonlocal nature of equations governing (conditional) expectations (ensemble means) of flow and transport variables in which fluxes depend not only on current local but also on past and distant gradients of heads and concentrations. We present approximate computational solutions to these equations which illustrate non-Darcian mean flow and anomalous mean transport in two-dimensional domains as functions of flow regime, proximity to boundaries and extent of conditioning on measured parameter values. We also show approximate solutions of corresponding conditional second moment equations which provide measures of uncertainty associated with smooth estimates of concentration and flux.